Interdisciplinary Applied Mathematics

The above formulation for two spheres has been extended to general curved hydrophobic surfaces by (Vinogradova, 1996). The results are very similar to those of the case of two spheres. For example, the resistance force is given by

рг = -ТГ7тГ‘(1013)

hlW 12

where f* is defined by the same expression of equation (10.12), but the geometry is now described by the curvatures of the two surfaces as follows:

11

12

1111

+ vtx + vw +

R— R+

R— R2 J

1

R—R+

+

1

R— R2

+ cos2ф

+ sin2ф

11

+

1

R+ R+

+

1

R R—

R— Ri

R+R-

Here R+ and R— denote the maximum and minimum principal radii of the surface, and thus I1 and I2 are the mean and Gaussian curvatures

of the effective surface, respectively. Also, ф defines the orientation of the two coordinate systems attached to the two surfaces. For example, we can consider the interaction of a sphere with a plane, a case typical in the surface force apparatus, in which case we obtain I1 = 1/R and I2 = 1/(4R2).

Similarly, we can model two crossed cylinders for which R+, R+ ^ ж and ф = п/2, so the two invariants are

Ii

2 U2~ + Ri

and I2

1

АЩВф ‘

The reader can try to determine the two invariants for the case of two